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It is often meaningful to obtain the partition coefficients of molecules by calculation. The molecular structure and extent of ionization are the primary factors in calculating the partition coefficient. The standard partition coefficient of ionized and unionized species calculated from the molecular structure is based largely on the atomic logP increments given in Viswanadhan et al. The extent of ionization at a given pH is obtained from the predicted pK_{a} of the molecule. Our calculation method takes into account the effect of the counterion concentration on logD and logP.
Throughout the document we use the following symbols:
- P_{i} (upper case) is the macro partition coefficient, where subscript i refers to the ionization state of species included in P_{i}, e.g. i= -2, -1, 0, +1, +2
- p_{i} (lower case) is the micro partition coefficient, where subscript i refers to the microspecies, e.g. i=1,2,3,4,…
- D is the distribution coefficient
- [ ] means concentration of microspecies
- logP is the logarithm of the partition coefficient
- logD is the logarithm of the distribution coefficient
The partition coefficient is the ratio of the concentration of the compound in octanol to its concentration in water. The distribution coefficient is the ratio of the sum of the concentrations of all species of the compound in octanol to the sum of the concentrations of all species of the compound in water. Based on acidic/basic dissociation reactions, we can introduce the concept of a partition coefficient for cationic and anionic species and for neutral species.
The following figure gives the definition of partition and distribution coefficients for ionized and unionized species.
Fig. 1 Partition and distribution coefficients for ionized and unionized species
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- Partition coefficient of the neutral species is
- Partition coefficients of the anionic and the cationic species are
and
- Distribution coefficient of the original molecule is
The micro partition coefficient is the ratio of the concentration of two microspecies defined with p_{i} as expressed with the next relation:
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Macro partition coefficients P_{0}…P_{i} can also be expressed as a function of micro partition coefficients p_{0}…p_{i}. From the definition of micro partition coefficients we obtain the following formula for the concentration of microspecies in octanol:
[microspecies]_{i,octanol} = p_{i} ⋅[microspecies]_{i,water}
where p_{i} is the micro partition coefficient of microspecies i.
For example, P_{-1} includes four micro partition coefficients (p_{1}, p_{2}, p_{3}, p_{4}). They are given by the following formulae:
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After substituting the p_{i}s into the original formula for P_{-1} we get the following simpler formula which includes only aqueous concentration of the appropriate microspecies:
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This can be further simplified if we introduce the acid dissociation constants of the A_{1}A_{2}B_{1}B_{2} molecule. The next five ionization reactions of the A_{1}A_{2}B_{1}B_{2} molecule are used to rearrange P_{-1} into a concentration free form:
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So we can further simplify the formula for P_{-1} to the following. This expression reveals that P_{-1} does not depend on the pH of the solution:
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Similarly, one could show that P_{0}, P_{+1}, P_{-2} and P_{+2} are also pH-independent.
In contrast to this, the distribution coefficient D does depend on the solution pH (see Klopman et al.):
LogP calculation methods
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Fig. 4 Homidium molecule
Ibuprofen has the typical logD vs. pH profile that is characteristic of acidic compounds. Lipophilic behavior of ibuprofen will be dominant when its carboxylic group is unionized (at low pH). At higher pH values the carboxylic group reaches the fully ionized state and hydrofilicity becomes more enhanced.
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Fig. 6 Ibuprofen and its logD-pH plot
The measured distribution coefficients also depends on the measurement method. The shake flask and the pH-metric methods are the most popular in practice. The figure below shows the calculated and measured logD of lignocaine as function of pH.
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Fig. 7 Lignocaine and its logD-pH plot
References
- Viswanadhan, V. N.; Ghose, A. K.; Revankar, G. R. and Robins, R. K., J.Chem.Inf.Comput.Sci., 1989, 29, 3, 163-172; doi
- Klopman, G.; Li, Ju-Yun.; Wang, S.; Dimayuga, M.: J.Chem.Inf.Comput.Sci., 1994, 34, 752; doi
- PhysProp^{©} database, webpage
- Csizmadia, F.; Tsantili-Kakoulidou, A.; Panderi, I. and Darvas, F., J.Pharm.Sci., 1997, 86, 7, 865-871; doi
- Bouchard, G.; Carrupt, P. A.; Testa, B.; Gobry, V. and Girault, H. H., Pharm.Res., 2001, 18, 5, 702-708; doi